More than once, evangelical friends tried to convert me used some version of Pascal's wager. Pascal argued that either God exists, or he does not; if he does exist, and you believe in him, you will receive an infinite reward in heaven; if you believe and you are wrong, you suffer no loss; on the other hand, if you do not believe and are right, your gain from a life of careless leisure is only finite. Therefore, he reasoned, balancing infinite gain on the one hand and finite gain on the other, it is logical to chose the infinite gain.

Pascal was one of the first to use a decision-making under uncertainty argument to justify belief in God. The argument's logical basis lies in decision theory: suppose you believe that God exists with probability p. Your expected utility from a belief in God, according to Pascal, is (infinity)*p+(some finite number, i.e. your utility if you were wrong)*(1-p)=infinity; your expected utility from non-belief is finite.

Putting aside a couple of obvious objections -- how can one assign probabilities to God's existence? -- the argument can be used to justify support in any deity and any religion -- does the idea of infinite gain make any sense at all -- Pascal's argument still suffers from a number of problems.

For one thing, the utility of eternal stay in heaven is not necessarily infinite. Sure, the gift is infinitely good; but your capacity to appreciate it is only finite.

Alternatively, if one accepts the idea of infinite utility, what about the possibility that a forgiving God will reward you with an eternity in heaven regardless? With two infinities on the balancing table, the argument does not hold.

What if you believe the probability of God's existence is zero? This does not mean that you believe God's existence is impossible. Plenty of events with probability zero occur all the time. Consider a dart thrown at a dartboard which hits any point on the dart board uniformly. What is the probability that dart will hit the point it actually hit? There are an infinite number of points and the probability of hitting any one point is zero; yet some point gets hit. If you believe that the truth of Christian doctrine is akin to the chance of a dart hitting a given point on the dartboard, it is logical to assign p=0 in which case the argument fails.

Finally, consider a thought experiment: rather than believe in God, you throw a fair coin, and believe in God if it falls on heads. What is your expected utility? It is (1/2)*(infinity)+(1/2)*(some finite number) = infinity. So this strategy is, by Pascal's argument, no worse than belief in God! Quoting from the Pascal entry in the Stanford Encyclopedia of Philosophy,

Suppose that you choose to ignore the Wager, and to go and have a hamburger instead. Still, you may well assign positive and finite probability to your winding up wagering for God nonetheless; and this probability multiplied by infinity again gives infinity. So ignoring the Wager and having a hamburger has the same expectation as outright wagering for God. Even worse, suppose that you focus all your energy into avoiding belief in God. Still, you may well assign positive and finite probability to your efforts failing, with the result that you wager for God nonetheless. In that case again, your expectation is infinite again. So even if rationality requires you to perform the act of maximum expected utility when there is one, here there isn't one. Rather, there is a many-way tie for first place, as it were.